This phase I/II trial studies the side effects and best dose of donor natural killer (NK) cell therapy and to see how well it works when given together with fludarabine phosphate, cyclophosphamide, total-body irradiation, donor bone marrow transplant, mycophenolate mofetil, and tacrolimus in treating patients with hematologic cancer. Giving chemotherapy, such as fludarabine phosphate and cyclophosphamide, and total-body irradiation before a donor bone marrow transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Giving an infusion of the donor's T cells (donor lymphocyte infusion) may help the patient's immune system see any remaining cancer cells as not belonging in the patient's body and destroy them (called graft-versus-tumor effect). Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells. Giving mycophenolate mofetil and tacrolimus after the transplant may stop this from happening.

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